Summary: A model for the exothermic oxidation of a porous solid reactant is considered. The effects of oxidant consumption are included in the model as well as are the effects of spatial variations in both temperature and oxidant concentration. The governing equations are made dimensionless so as to hightlight the ambient temperature \(T_ \alpha\), thus enabling critical values of \(T_ a\) to be identified clearly. Two types of boundary conditions are considered, Dirichlet boundary conditions, where the oxidant concentration and temperature are fixed at the surface of the solid, and Robin boundary conditions, where a finite heat and mass transfer is allowed between the surface of the solid and the surrounding reservoir. Bifurcation diagrams are obtained, giving plots of a representative (dimensionless) reactant temperature against the (dimensionless) ambient temperature, for both sets of boundary conditions. Two features are seen in these bifurcation diagrams, namely a hysteresis point, giving rise to multiple solution branches, and a point where the upper critical (extinction) point touches the zero ambient temperature axis, leading to the onset of disjoint solution branches. Both these features are considered further in detail.